Orthogonal frequency-division multiplexing (OFDM) is a method of encoding digital data on multiple carrier frequencies. OFDM has developed into a popular scheme for wideband digital communication, used in applications such as digital television and audio broadcasting, DSL Internet access, wireless networks, powerline networks, and 4G mobile communications.
For a sufficiently large number of tones, OFDM signal values are Gaussian, or normally, distributed, by the central limit theorem, which states that, given certain conditions, the arithmetic mean of a sufficiently large number of iterates of independent random variables, each with a well-defined expected value and well-defined variance, will be approximately normally distributed, regardless of the underlying distribution.
The Gaussian distributed signal components generate Rayleigh distributed signal amplitude values, Rayleigh distributions being continuous probability distributions for positive-valued random variables, specifically the distribution of the magnitude of a two-dimensional random vector whose coordinates are independent, identically distributed, mean 0 normal variables. In signals typical of the current state of the art, the Rayleigh distribution is typically heavy-tailed, i.e. a larger proportion of its population rests within its tail than would under a normal distribution. The long tail of the Rayleigh distribution, i.e. the portion of the distribution having a large number of occurrences far from the “head” or central part of the distribution, indicates the presence of large amplitude values, resulting from constructive interference. These large amplitude values can saturate the power amplifier, used in the processing chain, thus creating signal distortion which reduces demodulation performance.
This causes existing techniques for Orthogonal Frequency-Division Multiplexing (OFDM) to suffer from large peak-to-average power ratio (PAPR). The PAPR is the peak amplitude squared (giving the peak power) divided by the mean square value squared (giving the average power).
Many techniques have been developed to mitigate the PAPR problem, including signal companding. Companders are attractive because of their simplicity and effectiveness. Companders apply an amplitude weight to the signal, generally downweighting large amplitude values, while upweighting smaller values to maintain constant power. The large amplitude downweighting, while simultaneously maintaining the same power level, reduces the PAPR. The amplitude weighting introduces signal distortion, so it is important to design companders that introduce minimal distortion to limit demodulation errors, while also providing out-of-band power rejection.
Early solutions adapted companders from speech and audio coding, such as μ-law and A-law companders. Subsequent approaches decomposed the amplitude interval into disjoint regions over which different amplitude weighting functions were used. Modern companders are derived by transforming the Rayleigh amplitude distribution into a more favorable distribution to reduce PAPR. Among these transformations, piecewise transformations, i.e. a function composed of straight-line sections, have been used, either uniform, linear, nonlinear, or piecewise linear.
A need still exists, however, for solutions which minimally alter the original Rayleigh amplitude distribution.